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© Copyright 2000, Jim Loy
The excellent book, Self-Working Number Magic (subtitled "101 Foolproof Tricks") by Karl Fulves contains the following cute puzzle (a cryptarithm, much like a Word Arithmetic), with an incomplete solution. Every letter stands for a different digit:
EVE/DID=.TALKTALKTALKTALKTALKTALK...
You may want to work on this, before looking at the solutions.
The book's solution:
There are a number of ways (including Geometric Series) to show that 0.TALKTALKTALKTALKTALKTALK...=TALK/9999. So EVE/DID=TALK/9999. The author then leaps to the conclusion that DID must be a factor of 9999. 9999=3x3x11x101. The factors of 9999 of the form DID are 101, 303, and 909. Also we know that E<D. A great deal of experimenting with the other numbers shows that there is only one solution of this type:
242/303=.7986798679867986...
My solution:
Of course DID need not be a factor of 9999. It must be a factor of 9999xEVE. Much more experimenting with all of the numbers shows that there is a second solution:
212/606=.3498349834983498...
There are no other solutions.